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Properties

Label 16-468e8-1.1-c0e8-0-0
Degree 1616
Conductor 2.301×10212.301\times 10^{21}
Sign 11
Analytic cond. 8.85571×10−68.85571\times 10^{-6}
Root an. cond. 0.4832820.483282
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·13-s + 16-s + 4·37-s − 4·61-s − 4·73-s + 4·97-s − 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 4·208-s + 211-s + 223-s + ⋯
L(s)  = 1  − 4·13-s + 16-s + 4·37-s − 4·61-s − 4·73-s + 4·97-s − 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 4·208-s + 211-s + 223-s + ⋯

Functional equation

Λ(s)=((216⋅316⋅138)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((216⋅316⋅138)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 216⋅316⋅1382^{16} \cdot 3^{16} \cdot 13^{8}
Sign: 11
Analytic conductor: 8.85571×10−68.85571\times 10^{-6}
Root analytic conductor: 0.4832820.483282
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 216⋅316⋅138, ( :[0]8), 1)(16,\ 2^{16} \cdot 3^{16} \cdot 13^{8} ,\ ( \ : [0]^{8} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.22174682040.2217468204
L(12)L(\frac12) ≈\approx 0.22174682040.2217468204
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1−T4+T8 1 - T^{4} + T^{8}
3 1 1
13 (1+T+T2)4 ( 1 + T + T^{2} )^{4}
good5 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
7 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
11 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
17 (1+T4)2(1−T4+T8) ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )
19 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
23 (1−T+T2)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
29 (1+T4)2(1−T4+T8) ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )
31 (1+T4)4 ( 1 + T^{4} )^{4}
37 (1−T+T2)4(1+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T^{2} )^{4}
41 (1+T4)2(1−T4+T8) ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )
43 (1−T2+T4)4 ( 1 - T^{2} + T^{4} )^{4}
47 (1+T4)4 ( 1 + T^{4} )^{4}
53 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
59 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
61 (1+T)8(1−T+T2)4 ( 1 + T )^{8}( 1 - T + T^{2} )^{4}
67 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
71 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
73 (1+T+T2)4(1−T2+T4)2 ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}
79 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
83 (1+T4)4 ( 1 + T^{4} )^{4}
89 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
97 (1−T+T2)4(1−T2+T4)2 ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.03443068434407484399511134375, −4.97859846744122077983945342904, −4.93257593728763114415825579104, −4.77175819767595790785267465039, −4.62118637118897144266460594796, −4.34398802938013565083767026141, −4.28373941085487304598572801077, −4.28271590250825820772733275414, −4.06093059798509731247237750447, −3.95194803535709756617643671174, −3.66425450613093046692507366001, −3.50475593461122742915986499654, −3.38716822873934351229709467686, −2.98191306128880075412118192744, −2.86678889395622369762212537693, −2.80090024452565737098268707264, −2.79188106071294506308134056415, −2.56153115005259617070933379260, −2.53651263033368204614480316945, −2.14086036182289266783839130184, −2.04035531291014471096641922776, −1.60796657142899555613497392353, −1.45796747499801637898033505971, −1.39392086218897431406771882935, −0.841174426196479551750983974367, 0.841174426196479551750983974367, 1.39392086218897431406771882935, 1.45796747499801637898033505971, 1.60796657142899555613497392353, 2.04035531291014471096641922776, 2.14086036182289266783839130184, 2.53651263033368204614480316945, 2.56153115005259617070933379260, 2.79188106071294506308134056415, 2.80090024452565737098268707264, 2.86678889395622369762212537693, 2.98191306128880075412118192744, 3.38716822873934351229709467686, 3.50475593461122742915986499654, 3.66425450613093046692507366001, 3.95194803535709756617643671174, 4.06093059798509731247237750447, 4.28271590250825820772733275414, 4.28373941085487304598572801077, 4.34398802938013565083767026141, 4.62118637118897144266460594796, 4.77175819767595790785267465039, 4.93257593728763114415825579104, 4.97859846744122077983945342904, 5.03443068434407484399511134375

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.